Murray Park graduate Andrew Shinnie broke the deadlock for Caley Thistle in a first half which they were second best in at times.
And as frustration kicked in amongst the home ranks after the restart, Gary Warren’s header and a Graeme Shinnie penalty put the Highlanders into the semi-finals.
Gers had gone into the match with high hopes of becoming the first Third Division side to reach the last four of the competition.
The first three rounds, after all, had been kind to the Light Blues, with nine goals scored and none conceded as East Fife, Falkirk and Motherwell were beaten.
As it was, this was a step too far for them and they must now pick themselves up for this weekend’s Scottish Cup tie with Alloa after their heaviest loss of 2012/13 yet.
For the first time this season, boss McCoist retained the same starting line-up and substitutes from a previous game after Sunday’s 2-0 victory at Clyde.
And with almost 28,000 home fans cheering them on, the onus was on Rangers to begin brightly as they sought a place in the last four.
The hosts had the first chance of the game, a shot across goal and wide from Dean Shiels after he pounced on a defensive mistake.
In reply, Aaron Doran was on target with a well struck attempt which was on target and it needed two attempts from Neil Alexander before he held it.
The goalkeeper was called into action once more moments later as he got down really well to claw a firm Billy McKay drive around his near post after he broke the offside trap.
Back came Gers in a lively start, with Lee McCulloch laying off for Lewis Macleod to shoot and his effort was eventually smothered by away goalie Antonio Reguero.
That seemed to spur McCoist’s men on, with Andy Little bringing out a good stop from the Spaniard seconds later.
Macleod tried his luck again too but with the former Ferrol player beaten this time, he just cleared the woodwork too.
Rangers had had the better chances but Inverness arrived in Glasgow having scored in each of their 13 games so far this season – and they were to strike again here after 27 minutes.
It was a slack goal to concede too, with former Gers midfielder Andrew Shinnie getting on the end of a Doran pass to lift over Alexander with the back line caught out.
McCulloch tried to rouse his team-mates with a flashing header from Ian Black’s corner but it went wide.
Lee Wallace had a shot charged down as well and when the rebound fell to Little, it was always asking a lot for him to put a header in at an awkward height.
Chris Hegarty also went close with a nod on from a Wallace free kick which flew just over as the Light Blues piled plenty of pressure on without making it count.
Given the amount of chances they had had in the first half, Rangers were unlucky to trail by that single goal at the midway point.
And they kept charging forward in a bid to level at the start of the second, with McCulloch turning a rapid Little cross past the post.
With that, however, the tide seemed to turn back in favour of Inverness and Philip Roberts should have done so much better in front of goal than steer away from the target.
Although he was slack with his finish, Gary Warren was more ruthless with a header at a corner which sneaked over the line on the hour.
That knocked the stuffing out of McCoist’s side and although Kevin Kyle and Barrie McKay were thrown on, Inverness might have extended their lead further almost immediately.
With Hegarty short as he passed back to Alexander, it took an excellent stop to keep out Billy McKay when he pulled the trigger.
The Caley Jags forward had a chance to make amends for that a few minutes later but again Alexander was well placed to save as he enjoys a good season personally.
Butcher’s visitors would grab a third though as Anestis Argyriou, on for Ross Perry, was ruled to have fouled Andrew Shinnie in the box.
His brother Graeme stepped up to slide home from 12 yards after referee Steven McLean pointed somewhat harshly to the penalty spot and that capped a disappointing night.
RANGERS: Alexander; Hegarty, Perry (Argyriou 46), Emilson, Wallace; Black, Hutton (Kyle 63), Macleod; Little, McCulloch, Shiels (McKay 63).
SUBS: Gallacher, Aird.
BOOKED: Cribari, Black, Alexander.
INVERNESS CT: Reguero; Raven, Warren, Meekings, G Shinnie; Doran, Jones, Draper, Roberts (Sutherland 83); A Shinnie (Foran 88); McKay.
SUBS: Esson, Pepper, Cooper.
GOALS: A Shinnie 27, Warren, 60, G Shinnie pen 78.
BOOKED: Raven, McKay, A Shinnie.
REFEREE: Steven McLean.
